System Of Nonlinear Equations Research Articles

Extended Lambert-Based Approach for Ground-Track Adjustment with Specified Overflight Altitude

This paper presents an extended Lambert-based approach for ground-track adjustment under J2 perturbation, aiming to achieve a satellite flyby over a designated ground location at a predetermined altitude. Extended equations are developed in the orbital plane based on Lagrange’s form for the unperturbed Lambert’s problem to address the J2-perturbed two-body motion. By combining them with the out-of-plane flight time equation constraint regarding the ground-track adjustment, a nonlinear system of equations with two unknowns and two equations is established and split into different modes. To solve these modes, a modified Newton–Raphson method is used, and a necessary condition is established to determine the maximum allowable number of revolutions. To enhance computational efficiency, a radius restriction method is proposed to narrow down the allowable range of the number of revolutions. The proposed algorithm is applicable to various scenarios, including single- or multiple-day flights, ascending or descending overflights, coplanar or non-coplanar transfers, and initial or reverse-initial directional transfers. The simulation results demonstrate that the proposed algorithm is effective and accurate for the perturbed adjustment problem.

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar-Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar-Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilising ATDM.

Similarity solution for the magnetogasdynamic shock wave in a self-gravitating and rotating ideal gas under the influence of radiation heat flux

The Lie invariance method is used to analyze the one-dimensional, unsteady flow of a cylindrical shock wave in a rotating, self-gravitating, radiating ideal gas under the influence of an axial or azimuthal magnetic field, with an emphasis on adiabatic conditions. The analysis assumes a stationary environment just ahead of the shock wave and considers variations in fluid velocity, magnetic field, and density within the perturbed medium just behind the shock front. In the governing equations, the impact of thermal radiation under an optically thin limit is integrated into the energy equation. Utilizing the Lie invariance method, the set of partial differential equations governing the flow in this medium is transformed into a system of nonlinear ordinary differential equations (ODEs) using similarity variables. Two distinct cases of similarity solutions are obtained by selecting different values for the arbitrary constants associated with the generators. Among these cases, one yields similarity solutions assuming a power-law shock path and the other an exponential-law shock path. For both cases, the resulting set of nonlinear ODEs are numerically solved using the 4th-order Runge–Kutta method in MATLAB software. The article thoroughly explores the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), σ (ambient density exponent), l1 (rotational parameter), and G0 (gravitational parameter) on the flow properties. The findings are visually presented to offer a comprehensive insight into the effects of these parameters.

Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators

In the context of complete strong b-metric-like spaces, we prove new multi-fixed point solutions for the pair of multivalued, dominated operators that fulfill the generalized nonlinear contractions on a closed ball. We employ a mix of two different types of mappings in our approach: one is a class of multi-dominated mappings, while the other is a weaker class of strictly increasing mappings. Additionally, some new fixed-point results concerning the multi-graph-dominated structure in graph contraction are presented. To validate the hypothesis of acquired results, a few sample cases are presented. A numerical experiment has been carried out to approximate the fixed point. In addition, to demonstrate the originality of our findings, we proposed simple and efficient solutions to the system of fractional differential equations and nonlinear Volterra-type integral equations.

Implementation and Linearization of a Coupled Panel and Vortex Particle Method in State-Space Form

This paper describes the implementation and linearization of a coupled panel and vortex particle method in a state-variable form. More specifically, the coupled panel and vortex particle dynamics are formulated as a nonlinear system of ordinary differential equations in first-order form to be self-contained and inherently linearizable. Linearization of this coupled panel and vortex particle method is demonstrated via finite differencing and a novel analytical linearization technique to yield a linear time-invariant representation. The code is implemented in MATLAB® and validated against an open-source aerodynamic solver for a fixed wing in both stationary and unsteady conditions. The linearized models are verified against the nonlinear dynamics both in the time and frequency domains. Linearized models accurately represent the wake dynamics about the equilibrium condition for moderate amplitude inputs and for input frequencies covering the typical frequency range of flight dynamics and flight controls, i.e., 0.3–30 rad/s. Analytical linearization is shown to abate the cost of linearization over perturbation methods by O(n2), where n is the total number of states of the system. Linearized models of the coupled panel and vortex particle dynamics have applications in the flight dynamics of both fixed- and rotary-wing vehicles, where these dynamics can be used to augment the rigid-body dynamics. The coupled rigid-body and wake dynamics can be leveraged to assess the stability, response characteristics, and handling qualities of vehicles experiencing aerodynamic interactions between rotors, wings, and/or obstacles.

Sensitivity analysis and optimization of heat transfer in the time-reliant magnetohydrodynamic bioconvective flow of radiative couple-stress hybrid nanofluid through a squeezing channel with viscous dissipation: Entropy analysis

This research presents a statistical investigation of heat transmission in the time-reliant flow of blood-based hybrid nanofluid-containing microorganisms via a squeezing permeable channel utilizing a couple-stress non-Newtonian fluid model. Nanoparticles of SWCNT (single-wall carbon nanotubes) and MWCNT (multi-wall carbon nanotubes) are incorporated in the base liquid due to their low toxicity, special physio-chemical characteristics, and suitable surface modifications. In addition, chemical reactions, heat sources and sinks, radiation, viscous dissipation, and an external magnetic field impact the flow. The current numerical study stands out because it optimizes heat transmission rate using a face-centered central composite design framework with RSM (Response Surface Methodology) and analyses its sensitivity toward effective parameters. Examining the system's efficiency by evaluating the entropy produced in the flow process is another noteworthy part of this research. The complex system of interrelated non-linear partial differential equations is converted into a set of ordinary differential equations using suitable transformations. Later, solved via a semi-analytical technique, HAM (homotopy analysis method). According to the outcomes of the comparison, the hybrid nanofluid (blood/SWCNT-MWCNT) outperforms the nanofluid (blood/SWCNT) in terms of thermal performance. The rates of thermal energy transmission are very sensitive to the Eckert number and somewhat least affected by thermal radiation. Nanofluids and hybrid nanofluids have their heat transmission rates improved by including nanoparticles, heightened radiation, and magnetic effects. The system produces more entropy when the magnetic parameter and Eckert number are higher; however, it can be minimized by controlling the radiation effect. The results have important consequences for engineering industries, especially in the development and improvement of biomedical equipment and drug delivery systems, where accurate management heat transfer is crucial.

Parametrically excited shape distortion of a submillimeter bubble

The existence of finite amplitude shape distortion caused by parametrically excited surface instabilities for a gas bubble in water driven by a temporally periodic, spatially uniform pressure field in an axisymmetric geometry is investigated. Employing a nonlinear coupled system of equations which includes shape mode interactions to third order, the resultant spherical oscillations, translation, and shape distortion of the bubble are modelled, placing no restriction on the size of the spherical oscillations. The model accounts for viscous and thermal damping with compressibility effects. The existence of synchronous and higher order parametrically induced sustained, finite amplitude, periodic shape deformation is demonstrated. The excitement of an odd shape mode via the synchronous mechanism is shown to give rise to linear bubble self-propulsion. For larger driving amplitudes, it is shown that more than one shape mode can be parametrically excited at the same driving frequency but by different resonance mechanisms, leading to more involved shape deformation and the increased possibility of bubble self-propulsion.

Low-Computational-Complexity Zeroing Neural Network Model for Solving Systems of Dynamic Nonlinear Equations

Low-Computational-Complexity Zeroing Neural Network Model for Solving Systems of Dynamic Nonlinear Equations

Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

Converting estimated breeding values from the observed to probability scale for health traits

Dairy cattle health traits are paramount from a welfare and economic viewpoint; therefore, modern breeding programs prioritize the genetic improvement of these traits. Estimated breeding values for health traits are published as the probability of animals staying healthy. They are obtained using threshold models, which assume that the observed binary phenotype (i.e., healthy or sick) is dictated by an underlying normally distributed liability exceeding or not a threshold. This methodology requires significant computing time and faces convergence challenges as it implies a nonlinear system of equations. Linear models have more straightforward computations and provide a robust approximation to threshold models; thus, they could be used to overcome the mentioned challenges. However, linear models yield estimated breeding values on the observed scale, requiring an approximation to the liability scale analogous to that from threshold models to later obtain the estimated breeding values on the probability scale. In addition, the robustness of the approximation of linear to threshold models depends on the amount of information and the incidence of the trait, with extreme incidence (i.e., ≤ 5%) deviating from optimal approximation. Our objective was to test a transformation from the observed to the liability and then to the probability scale in the genetic evaluation of health traits with moderate and very low (extreme) incidence. Data comprised displaced abomasum (5.1M), ketosis (3.6M), lameness (5M), and mastitis (6.3M) records from a Holstein population with a pedigree of 6M animals, of which 1.7M were genotyped. Univariate threshold and linear models were performed to predict breeding values. The agreement between estimated breeding values on the probability scale derived from threshold and linear models was assessed using Spearman rank correlations and comparison of estimated breeding values distributions. Correlations were at least 0.95, and estimated breeding value distributions almost entirely overlapped for all the traits but displaced abomasum, the trait with the lowest incidence (2%). Computing time was ∼3x longer for threshold than for linear models. In this Holstein population, the approximation was suboptimal for a trait with extreme incidence (2%). However, when the incidence was ≥6%, the approximation was robust, and its use is recommended along with linear models for analyzing categorical traits in large populations to ease the computational burden.

Finite Difference with Quintic B-Splines for Solving a system of nonlinear Volterra Integro-Differential Equations of integer order

This study presents a novel numerical approach for solving systems of nonlinear Volterra Integro-Differential Equations (VIDEs) of integer order by integrating the Finite Difference Method (FDM) with Quintic B-Splines for. The proposed method aims to address the challenges posed by the nonlinearity and integral terms inherent in VIDEs while providing enhanced accuracy and stability in the numerical solution. An error estimate of the approximate solution is proved. Finally, Some example are presented to illustrate the simplicity and the effectiveness of the propose method

On the Role of Early Case Detection and Treatment Failure in Controlling Tuberculosis Transmission: A Mathematical Modeling Study

Tuberculosis (TB) remains a pressing global health concern, demanding urgent attention to mitigate its spread and impact. In this study, we present a rigorous mathematical model of TB transmission that incorporates early case detection and addresses the critical issue of treatment failure. Through the development of a system of nonlinear ordinary differential equations, we conduct comprehensive analyses to assess the dynamics of TB transmission and the efficacy of intervention strategies. Our findings underscore the urgent need for effective TB control measures. Mathematical analyses reveal that the model exhibits a TB-free equilibrium, which is globally asymptotically stable only if the control reproduction number falls below one. However, we identify a concerning phenomenon: the model demonstrates a forward bifurcation when the control reproduction number equals one, suggesting that the disease-free equilibrium loses its stability, while simultaneously, the stable unique endemic equilibrium begins to emerge. Moreover, sensitivity analysis highlights the complex interplay between case detection rates, treatment failure probabilities, and TB transmission dynamics. Contrary to expectations, increasing case detection rates and minimizing treatment failure probabilities may not consistently reduce the basic reproduction number or the size of the infected population. Instead, there exists a critical threshold for intervention effectiveness, beyond which TB transmission can be significantly curtailed. Biologically, this phenomenon may occur if there is no balance between case detection and treatment efforts. If treatment quality does not improve, then case detection will not have a significant impact, and in the worst case scenario, it can exacerbate the intervention’s negative effects. These findings underscore the urgency of implementing targeted intervention strategies to combat TB transmission effectively. Failure to meet the critical intervention threshold risks undermining TB elimination efforts and exacerbating the global TB burden. Through numerical simulations, we elucidate potential intervention scenarios necessary for achieving TB elimination goals in human populations. In conclusion, our study highlights the urgent imperative for coordinated action to control TB transmission effectively. By elucidating the dynamics of TB spread and intervention efficacy, we provide valuable insights to inform evidence-based policy decisions and accelerate progress towards TB elimination on a global scale.

Determining the influence of wheelset arrangement in the model 18-100 bogies on the level of steering efforts in the wheel-rail flange contacts

The object of this study is the process of directing the bogies of model 18-100 freight cars along a rail track, in particular, along curved sections of the track. The task to be solved was to determine the influence of wheelset arrangement on the level of steering forces in the flange contacts. A calculation diagram and a mathematical model of fitting the bogie into the curved section of the track have been built. The bogie loading scheme by external forces, including lateral rocking forces acting on the car in a curve, has been refined. In this case, the method of pseudo-statics of the mechanical system, which is a system of nonlinear algebraic equations, was applied. The calculation module Given-Find in the Mathcad software package (USA) was used to solve the mathematical model. It was established that the misalignment of wheelsets in the frames of model 18-100 bogies was of an accumulative nature. At the maximum operating angles of misalignment of wheelsets, the lateral steering forces in the flange contacts increase by 40–60 % compared to the rated setting. These angles can be up to 0.015 rad (0.85 degrees). The field of practical application of the results is railroad transportation, in particular, the system of maintenance and repair of freight cars on bogies of the 18-100 model. At the same time, the condition for the practical application of the research results is the expediency of introducing into the maintenance system the technological operation of controlling the deviation of wheelset arrangement in the bogie relative to the rated one. The current study will contribute to the construction of a measuring system for monitoring the deviation of wheelset arrangement in the bogie relative to the rated one. This proves the expediency of introducing into the trolley maintenance system the technological operation of controlling the deviation of wheelsets and designing a device for monitoring this parameter

Research on acoustic localization method for multiple leakage sources of water-cooling wall in furnace

Abstract The water-cooling wall of the furnace often leaks under harsh conditions such as high temperature and pressure, which affects the safety and economic operation of the boiler. To achieve the localization of multiple leakage sources in water-cooling wall, an acoustic positioning method based on improved differential evolution and density-based spatial clustering of application with noise (IDE-DBSCAN) is proposed. The nonlinear equation system is constructed using time delay of arrival (TDOA) obtained by the generalized second cross-correlation algorithm, and the IDE algorithm is used to solve nonlinear equation system and obtain multiple spatial positions corresponding to multiple TDOA sequences. To eliminate false leakage sources, the paper presents IDE-DBSCAN to cluster the multiple estimated positions. The number of clusters obtained is the number of leakage sources, and the cluster centers are the location coordinates of the leakage sources. The numerical simulation results show that compared to traditional acoustic positioning methods, IDE-DBSCAN has higher positioning accuracy and better anti-noise ability. In addition, experiments are implemented on multiple leakage sound sources within a measurement area of 1m×1m×1.06m, and the positioning errors remain within 5cm. The proposed method provides a theoretical basis and experimental verification for further research on multiple leakages localization of water-cooling wall in furnace.

Flow analysis of temperature-dependent variable viscosity Phan Thien Tanner fluid thin film over a horizontally moving heated plate

Comprehending the temperature-dependent variable viscosity Phan Thien Tanner fluid film flow over a horizontal heated plate with surface tension is essential to enhancing coating and lubrication process prediction models. This paper accords with the flow analysis of a temperature-dependent variable viscosity Phan Thien Tanner fluid film over a horizontal heated plate. The flow over a heated plate occurs as a result of the plate’s motion and the surface tension gradient. The Adomian decomposition method is utilized to solve a system of linear and nonlinear ordinary differential equations, resulting in series-form solutions. The series-form expressions for flow variables such as velocity, temperature, volume flow rate, and surface tension are derived. Moreover, the positions of stationary points are computed using MATHEMATICA. The analysis delineated that when the inverse capillary number, variable viscosity parameter, Deborah number, elongation parameter, and Brinkman number increase, the stationary points move closer to the heated plate. The temperature also rises with an increase in these parameters. The temperature rises with increasing viscous dissipation while it lowers with increasing thermal diffusion. When the Deborah number is high, the Phan Thien Tanner fluid behaves like a solid and the flow is only driven by the motion of the plate. A comparison between Newtonian and Phan Thien Tanner fluids is made for velocity, temperature, and stationary points as a special case.

Time Optimal and PID Controller for Armed Manipulator Robots

This paper is fourfold. First, three different time-optimal control problems for simulating the manipulator robots with one, two, and three arms are mathematically formulated. Corresponding to the related dynamical systems, the nonlinear system of ordinary differential equations is derived. It has been found that these problems are time-free and have distinct initial and boundary conditions, making them hard to solve. To find the minimum time with different payloads, a successful numerical method based on the finite difference and the three-stage Lobatto formula is proposed. Secondly, the related torquecontrolling problems are simulated, and then for one, two, and three armed manipulator robots, they are solved using the PID controller method. Thirdly, it is shown that, compared to the time-optimal controlling problem, the optimal PID torque controller solution takes more time to do the job than was expected. However, the solution in the PID controller method shows less oscillation than the time-optimal control problem. Fourthly, mathematical theories are used, and the numerical results for both methods and different payloads are compared.

Computational investigation of methanol-based hybrid nanofluid flow over a stretching cylinder with Cattaneo–Christov heat flux

Abstract This study investigates heat transfer rates in (AA7075-AA7072/Methanol) hybrid nanofluid flows, considering non-uniform heat sources and Cattaneo–Christov heat flux, with significant implications for aerospace engineering by enhancing thermal management in aircraft engines. The findings could revolutionize automotive cooling system efficiency, optimize heat dissipation in electronic devices, and advance the design of renewable energy systems such as concentrated solar power plants. The study aims to conduct a comparative analysis of (AA7075/Methanol) nanofluid and (AA7075-AA7072/Methanol) hybrid nanofluid flow, examining heat transfer rates, non-uniform heat sources, and Cattaneo–Christov heat flux theory around a stretching cylinder. Thermal radiation and the Biot number are also evaluated. Two different nanoparticles, AA7072 and AA7075, are used with methanol to create AA7075/Methanol nanofluid and AA7075-AA7072/Methanol hybrid nanofluid. The study compresses the resultant non-linear partial differential equation system and applies suitable similarity transformations to reduce the governing partial differential equations with boundary conditions to dimensionless form. The BVP4C shooting method in MATLAB is employed to numerically and graphically solve these dimensionless ordinary differential equations. The results indicate that higher curvature parameter values correlate with increased velocity and temperature distribution profiles. A rise in nanoparticle volume fraction reduces the radial velocity profile but increases the temperature profile. Temperature distribution profiles increase with higher thermal radiation parameter and Biot number values, while higher thermal relaxation parameter values decrease temperature. Additionally, thermal distribution profiles rise with increasing values of both the time-dependent heat source constant and space-dependent heat source parameter.

К выбору параметров регулятора непосредственного понижающего преобразователя с одноконтурной системой управления с учетом динамических нелинейностей

The article is devoted to the issues of choosing the parameters of a proportional-integral-differentiating regulator of a single loop control system with a buck converter taking into account dynamic nonlinearities This problem arises when it is necessary to design pulsed voltage conver¬ters with increased performance, when it is necessary to increase the cutoff frequency of an open-loop control loop without the possibility of increasing the carrier frequency of pulse width modulation due to limitations of the element base, which in some cases leads to the appearance of nonlinear effects. This must be taken into account when choosing both the open-loop cutoff frequency and the stability margin. The article proposes a nonlinear dynamic model of a direct buck converter with a proportional-integral-differentiating regulator both in the form of a piecewise smooth system of differential equations and in the form of a Poincaré map. A method for selecting controller parameters is proposed for a given open-loop cutoff frequency, and a given stability margin, as well as for given ranges of input voltage and load resistance based on the use of numerical methods for solving systems of nonlinear equations with given restrictions. The nonlinear dynamics of the system was studied for various sets of controller parameters. It is shown that with an increase in the cutoff frequency and small stability margins, undesirable dynamic modes may arise in the system when the filter capacitance drops within the calculated values, while linear dynamic models show the stability of the system. To eliminate the possibility of the emergence of undesirable modes, it is necessary to increase the stability margin with subsequent monitoring of the results using a nonlinear dynamic model. The proposed method for selecting the cutoff frequency of an open loop and the phase stability margin, based on the analysis of nonlinear dyna¬mics, allows you to design power supplies with increased performance.

Talbot Effect for the Manakov System on the Torus

In this paper, the Talbot effect for the multi-component linear and nonlinear systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles is investigated. Firstly, for a class of two-component linear systems satisfying the dispersive quantization conditions, we discuss the fractal solutions at irrational times. Next, the investigation to nonlinear regime is extended, we prove that, for the concrete example of the Manakov system, the solutions of the corresponding periodic initial-boundary value problem subject to initial data of bounded variation are continuous but nowhere differentiable fractal-like curve with Minkowski dimension 3/2 at irrational times. Finally, numerical experiments for the periodic initial-boundary value problem of the Manakov system, are used to justify how such effects persist into the multi-component nonlinear regime. Furthermore, it is shown in the nonlinear multi-component regime that the interplay of different components may induce subtle different qualitative profile between the jump discontinuities, especially in the case that two nonlinearly coupled components start with different initial profile.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.